Textures are visible in Texture viewport mode, but disappear in Render and Texture Paint
Make Sure your 3D Mesh is UV unwrapped.

Your material must have a texture attached.

As a temporary investigation tool you may select  ... Material Shading [Shadeless] or add more lights to the scene.

Here is an example.



We are in the texture pane ... indicated with red and white checkerboard icon.
In the image above you see Object Name, Material Name, Texture Name ... next to the push pin. Reading that line verifies the attributes are linked. These names indicate an active hierarchy. Object with subordinate Material with subordinate texture.  They are linked, attached, bound.  

It can be confusing that in 3D View Window you may see an image temporarily on your mesh.  Feature perhaps. During render only the texture pane setting will be rendered in Blender Render mode.  

Because I created the object, then created one material while the object was selected, the material is bound to the object.  Next I switched to the texture pane. I created a  new texture immediately after viewing the material.  The check box for the texture is selected.  The texture is bound to the material.

I am simply restating the settings you see here for emphasis.  Type [Image or Movie].  Image source you see a filename from the hard disk. By clicking the button you will see a select file dialog. 

More texture settings.



Mapping UV with understood default name. Influence [Color]